AI claim knocks a famous math conjecture off balance
Mathematician Levent Alpöge said Anthropic’s Fable 5 found a counterexample to the Jacobian conjecture, a 90-year problem in algebraic geometry.
By Georgia Hale · Staff Writer
3 min read
Mathematician Levent Alpöge said Sunday on X that he had used Anthropic’s Fable 5 model to disprove the Jacobian conjecture, a problem that has sat unresolved in algebraic geometry for almost 90 years.
The claim centers on a counterexample: one case that, if valid, would show the conjecture cannot be true in its stated form. The Jacobian conjecture was one of “Smale’s problems,” the list of major unsolved mathematics questions proposed by mathematician Stephen Smale in 1998.
Alpöge is a member of Harvard’s Society of Fellows, and his LinkedIn profile lists an affiliation with Anthropic. Mashable reported that it contacted Anthropic with questions about his work.
Fable 5 is Anthropic’s latest frontier AI model, according to Mashable. The outlet described it as the public version of Claude Mythos Preview, a model Anthropic has said had advanced cybersecurity capabilities that made it too dangerous for public release.
A big result, with a catch
Andrew Blumberg, a mathematician and professor with a joint appointment in mathematics and computer science at Columbia University, told Mashable the announcement did not change his view of what current AI systems can do.
“This did not cause me to update my priors about what AI can and can't do,” Blumberg said. “This is exactly the kind of thing I would expect AI to be able to do. If there was a counterexample that was concise and easy to state that people haven't found because it's a pain to search through all this stuff, AI will find it.”
Blumberg is involved in the First Proof project, which tests frontier large language models on research-level mathematics, according to Mashable.
He said a counterexample to the Jacobian conjecture would still count as a meaningful AI achievement in mathematics. His caution is about what kind of achievement it is.
Blumberg drew a distinction between proving a major theorem and finding a single object that breaks a conjecture. A proof can reveal why a mathematical claim is true. A counterexample can end the question without giving much new structure to study.
“Suppose that Moses came down from the mountain with tablets, and on the tablet was written, ‘Cancer can be cured.’ Would you care?” Blumberg told Mashable. “You don't just want the answer to the question. You want to learn something from the answer.”
He said Smale considered the Jacobian conjecture important because solving it might have clarified deeper mathematical structure.
“This counterexample tells us essentially nothing,” Blumberg said. “It's just, you know, there are a lot of polynomials, and it's hard for people to check them all, but it's not hard for machines.”
AI has been hunting conjectures
The Jacobian conjecture is not the only high-end math target that AI has recently taken on. OpenAI announced in May that an internal model had disproved the Erdős unit distance conjecture, a major question in discrete geometry.
Blumberg told Mashable that case was more useful because the AI did more than produce a lone counterexample. He said experts examined the disproof and used its ideas for additional work.
On the Jacobian claim, Blumberg said he was not enough of a specialist to rule out that the counterexample’s structure might contain useful lessons. Taken on its own, though, he said, “it's not in and of itself interesting.”
This story draws on original reporting from Mashable.